Antenna theory: Amplitude Variations Along a Ray
Power decreases in a general ray as the distance from the source increases. If we expand the constant-phase surface (eikonal) about the ray in a Taylor series, we obtain a surface described by its radii of curvature. The maximum and minimum values lie in the orthogonal principal planes. These radii of curvature determine the
amplitude spread of the wave from point to point on the ray. We compute the ratio of differential areas about the ray at two locations as
 (1)
where ρ1 and ρ2 are the principal radii of curvature and d is the distance between two points on the ray (Figure 1). The electric field variation along the ray becomes
 (2)
for the astigmatic ray spreading from unequal radii of curvature. When d = −ρ1 or d = −ρ2, GO fails because it predicts an infinite power density. We call these locations caustics. Remember that the ray always has differential area and never has any real area as implied by Figure 1. We have three special cases of the astigmatic ray:
 (3)
 (4) (5)The plane wave does not spread but has constant amplitude as distance changes. Both cylindrical and plane waves require infinite power, and they are therefore nonphysical, but we find them convenient mathematically.
FIGURE 1 Astigmatic ray. |
